Many of the best probabilistic models represent probability distributions only implicitly
The probabilistic models that represent distributions implicitly are, for example, the GANs. (Goodfellow is one of the authors of the original GAN model).
In the paper Variational Inference using Implicit Distributions (2017), the authors write

Implicit distributions are probability models whose probability density function may be intractable, but there is a way to

sample from them exactly and/or calculate and approximate expectations under them, and

calculate or estimate gradients of such expectations with respect to model parameters.


Implicit models have been successfully applied to generative modelling in generative adversarial networks (GANs)
A popular example of implicit models are stochastic generative networks: samples from a simple distribution - such as uniform or Gaussian - are transformed nonlinearly and non-invertably by a deep neural network

They also provide a table (table 1) that shows some probabilistic models that use an implicit distribution. Here $I$ denotes inference only. VI stands for variational inference.

See this blog post Variational Inference using Implicit Models, Part I: Bayesian Logistic Regression. See also the paper Learning in Implicit Generative Models (2017).